PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number 2, June 1972 HOMOMORPHISMS OF RINGS OF GERMS OF ANALYTIC FUNCTIONS WILLIAM R. ZAME1 Abstract. Let 5 and S' be complex analytic manifolds with 5 Stein. Let A^Sand X'^S' be compact sets with A'holomorphically convex. Denote by £(X) (respectively G{X')) the ring of germs on X (respectively A') of functions analytic near X (respectively X'). It is shown that each nonzero homomorphism of Q0C) into C(X') is given by composition with an analytic map defined in a neighbor- hood of X' and taking values in S. If S and S' are complex analytic manifolds, then every analytic mapping of S into S' induces (via composition) a homomorphism of the ring of analytic functions on S' into the ring of analytic functions on 5. It is an important and deep result that the converse is also true, providing that S is a Stein manifold (see [1]). In this note we obtain an analogous result for homomorphisms of rings of germs of analytic functions on compact subsets of a complex analytic manifold. We show that each such homo- morphism is given by composition with an analytic mapping. 1. Preliminaries and notation. Let S be a Stein manifold and i/an open subset of 5. We denote by 0(U) the ring of analytic functions on U. It is well known (see [1], for example) that each nonzero complex-valued homomorphism of C;iU) is continuous with respect to the topology on C'iU) of uniform convergence on compact subsets of U. We denote the space of such homomorphisms by AO(U) Iffe&iU) then/ denotes the function on Aß(U) defined by/ (a) = a(/) for each a in A@(U). Since 5 is Stein, AC(S)=S, so we have the natural restriction map ■7rrj:A<P(U)-^S given by 7rC7(a)(/) = a(/|C/). Rossi [4] has shown that A@iU) admits the structure of a Stein manifold in such a way that: (i) the evaluation map U^-Aß(U) is a biholomorphism of U with an open subset of A€iU) (we will regard U as an open subset of A@iU)); (ii) if fe&(U) then/ is the unique analytic extension off to AGiU) (so that <9(A<B(U))=Q(U))\ (iii) tttj is locally a biholomorphism. Received by the editors September 2, 1971. AMS 1970 subject classifications. Primary 32E25, 46E25. Key words and phrases. Germs of analytic functions, holomorphically convex sets. 1 Supported by National Science Foundation Grant GP-19011. ($j American Mathematical Society 197 410 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use